Keywords: epidemic, spatial models, cellular automata.
CR Classification: I.6.5, J.3.
Public health issues are seeing greater visibility in the media; a particular concern is viral spread through populated areas. Decreased worker productivity as a result of viral illness costs industry millions of dollars every year [1]. That is why it is desirable to model and predict the pattern of viral infection through a population. It is hoped that such models will assist in the prevention of epidemics.
There exist many models of epidemic spread, each with its own approach and set of assumptions. However, these models all share one property: the virtual world in which they run is an idealised one where noise and imperfections are filtered out. This arises from the difficulty to incorporate all the variables we see in nature into a simulation that has a reasonable execution time - hours rather than days. When modelling a complex system there is a trade off between a model's degree of abstraction and its usefulness; that is, without devaluing the results a model provides, which details can be left out?
Instead of abstracting a new model from scratch, I will examine previously proposed models and see what levels of detail they have adopted and proceed to create my own composite model. The majority of past approaches have used ordinary and partial differential equations (ODE's and PDE's). I examine those as well as mean field type (MFT) approximations [2] and cellular automata (CA). I differentiate between a modelling approach and a model in the sense that after adopting a particular approach, the inclusion of different virus spread parameters will produce different models.
After examining past models, evidence suggests that the CA approach provides usable models of epidemic spread by allowing realism properties such as heterogeneity in population densities to be easily integrated.
The focus of this modelling exercise is on the spatial behaviour of epidemics rather than absolute numbers and densities of infected individuals. In order to compare the relative usefulness of modelling approaches a set of standard criteria is examined. These criteria are based on the following factors that affect epidemic spread [3]:
All of the above will influence whether an infection will rapidly propagate through a population or head into extinction. Different models will have different assumptions regarding the above parameters and hence have different success rates in mimicking nature. The rest of this paper examines three modelling methodologies: ODE's and PDE's, MFT approximations, and CA to contrast how each approach incorporates the above listed epidemic spread parameters. A model's omission of a parameter does not imply the model is unrealistic, though it might mean that the model is designed to investigate the impact of one particular parameter independently of the others.
Deterministic approaches to modelling, such as those using ODE's and PDE's, are poor at representing small populations compared to probabilistic models such as CA [4]. They are poor in the sense that the results from ODE's diverge to unreasonable values as population size is scaled down towards zero. This divergence is because most ODE models make a number of simplifying assumptions:
Most of the above assumptions arise from regarding susceptible populations as continuous entities rather than comprising discrete individuals. By recognising that the spatial behaviour of an epidemic is ``strongly linked to the short range character of the infection process" as stated by Boccara [5], continuous differential equation models that neglect the individual are probably going to be misleading on all scales, not just small.
However, the continuous nature of ODE approximation is shown to be well suited to dealing with large populations [4], that is, in situations where localised correlations between individuals become negligible compared to the epidemic's macroscopic behaviour. The local correlations are lost because individuals are able to roam all over the world. This spatial mixing effect is introduced into PDE model's through a diffusion term and becomes a function of inputted initial and boundary value conditions.
The need for homogeneity before an ODE model can be drawn up means that the natural progression of an epidemic is not represented accurately. Variations in world population densities, variations in immunity and susceptibility levels, and variations in incubation and sickness time are all attributes of natural epidemics but are omitted in ODE simulations. Given that many spatial properties of epidemics are not realised by differential equation models, I choose not to use this approach when drawing up my own simulations.
Closely related to ODE and PDE models are those using MFT approximation; they also share similarities with CA. It is logical to hope that MFT approximations encapsulate the advantages of both deterministic and probabilistic models but it turns out that MFT approximations include differential equations [2] which is why I have classified the MFT approach as deterministic.
MFT approximations ignore localised correlations, which makes them similar to differential equation models [6]. MFT models assume that susceptible population density is uniform over the world, which is also similar to ODE/PDE models. Finally, MFT and ODE/PDE models share the assumption that hosts are capable of diffusing around the world, that is, the population is well mixed. Despite all these similarities MFT approximations are significantly different from differential equation models because their mixing parameter is a probabilistic variable [2]. Unlike ODE's where either all individuals diffuse or none at all, the decision to move around the MFT world is independent among individuals.
Similar to CA, MFT approximations utilize a lattice structure to emulate the spatial nature of epidemic spread. Each lattice site contains an individual who can exist in one of several states. The set of possible existence states is determined by the epidemic model being used. Many groups [5,6,7] have used MFT approximations as a point of comparison with the CA models they develop; particularly models investigating the effect of host motion. Noting that MFT approximations neglect localised correlations it appears meaningless to compare two disjoint approaches; CA models focus on the contact between individuals. But as Kleczkowski [2] describes, MFT and CA models converge when the MFT mixing parameter tends to infinity, that is, when the world contains more disorder than correlation. This convergence is analogous to the situation where differential equation and CA models converge when population size tends to infinity.
In the context of exhibiting realistic spatial behaviour, MFT approximations are only marginally better than modelling with differential equations. This is because they at least manage to portray some of the stochastic fluctuations as observed in nature by probabilistically determining host movement.
It is the spatial behaviour of epidemics that interests us and CA appears to fit well to that task. Epidemic spread in nature is a stochastic process so it seems logical to use a model that is probabilistic. According to Ahmed et al. and others [8,7], ``CA has a significant role in epidemic modelling since it can be shown that it is more general that ordinary and partial differential equations." This section, rather than discuss the suitability of CA as an approach, will examine existing CA models and identify which virus spread parameters should be incorporated into my composite epidemic model.
All of the following models are based on the SIR model superposed with CA. The letters in SIR correspond to the states an individual in a population can exist in: susceptible, infective and recovered [9]. Susceptible individuals, or susceptibles, can contract the pathogen from infectives who are already infected and who can later recover from the infection. There are variants of this model that introduce other intermediate states, for example, SEIR where `E' represents exposed individuals.
| Criteria | [7] | [5] | [4] | [9] | [6] |
| Wrap around world | × | × | × | × | × |
| Variable population size | × | ||||
| Uneven population density | × | × | × | ||
| Movement of hosts | × | × | × | ||
| Immunity after recovery | × | ||||
| Variable susceptibility | × | ||||
| Includes incubation time | × | ||||
| Includes latency time | × |
Factors such as population density, susceptibility and immunity, transmissibility, and infection times are parameters whose value directly affects the realism of the simulations we generate. They are parameters that add heterogeneity to the idealistic world that designers start building models from. Table 1 shows which parameters each of the examined models have chosen to implement.
In nature the population within a region is always changing. Internal events such as births and deaths increase and decrease the population respectively. External factors such as immigration and emigration have similar effects. Epidemic spread is affected by this constant flux in susceptible hosts but very few models include this flux. The reason might lie in the difficulty to implement such features or that most models have very specific applications where population variation is considered negligible.
Of the five groups mentioned here only one, Boccara et al. [6] has chosen to model population changes. They chose to include the death and birth rates of susceptibles and infectives to demonstrate how these parameters affect the stability of the endemic states that infected populations reach as time tends to infinity. Other models [5,4,9] focus on the movement and the heterogeneity of susceptibility in populations and abstract out the population size parameter.
Variations in population densities are implemented in CA models by allowing cells to remain empty [8]. This means a particular individual may have no neighbours to contract the infection from or pass the infection to. This is analogous to the isolation procedures used in real life to contain outbreaks. As seen in Table 1, most models implement this factor.
Susceptibility, immunity and transmissibility relate to how easily a contagion can pass between hosts. Probabilistic models such as the CA/SIR model represent high susceptibility, low immunity and high transmissibility by assigning high probabilities of infection when susceptibles come in contact with the contagion. These parameters are easily adjusted in CA models.
The length of time between an individual being infected and an individual showing signs of disease is known as the incubation time. The time lapse between being infected and becoming infective is known as latency. Both of these quantities are modelled by Ahmed et al. [9] though they do not discuss the significance of these times. Others fail to include these quantities in their models suggesting that these times do no have much impact on an epidemic's spatial behaviour.
Before a complex system can be modelled, consideration must be given to the scale and degree of detail the model will adopt. Substantial research has been done in epidemic modelling and many models have been devised. It makes sense to develop a new model by building upon and improving existing models rather than starting from scratch. Examining past works gives an indication on how much abstraction is required to make models useable and still preserve accuracy.
Several modelling paradigms have been examined to identify which factors have the biggest influence on epidemic spread. Differential equation models are well suited to examining large population effects but rely upon homogeneous environmental conditions. Mean field type approximations are used as benchmark functions to compare other models. Cellular automata models are well suited to mimicking the spatial pattern of epidemic spread. All these approaches have been used, but CA models prove to be the most effective for spatial applications.
Previous works [8,6,2,9,4,5,7] show that a variety of realism parameters can be easily incorporated into cellular automata models. Such parameters include those in Table 1. It is now left to take this composite list of parameters and develop a model that will effectively simulate epidemic spread as it is found in nature.